# Lesson 5

What is an Angle?

## Warm-up: Notice and Wonder: A Wall of Clocks (5 minutes)

### Narrative

The purpose of this warm-up is to draw students’ attention to the figures formed by pairs of segments that are joined at a point in preparation for an exploration of angles. Students may notice and wonder many things about the clocks, but describing how the figures formed by the hands and how they are the same and different are the important discussion points.

### Launch

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• 1 minute: partner discussion
• Share and record responses.

### Student Facing

What do you notice? What do you wonder?

### Activity Synthesis

• “Besides the time, what else changes when the long and short hands of the clocks turn?” (The directions of the hands, the figures created by the two hands, the appearance of the clock.)
• “Some of you may wonder what time and clocks have to do with lines, points, rays, and segments. Let’s keep thinking about this as we work on our first activity.”

## Activity 1: Tricky Figures (15 minutes)

### Narrative

In this activity, students work with a partner to replicate images of angles. One partner describes the figure and the other draws based on the verbal descriptions. The purpose of the activity is to draw students’ attention to how they use the vocabulary they have learned from previous lessons to describe the figures (MP6). In this synthesis, students learn that an angle is a geometric figure that is made up of two rays that share the same endpoint. Students may also become aware that they need a clear way to describe the size of the figure. Angle measurement is not addressed in this lesson, but in the process of describing or drawing the figures, students are likely to use terms such as “narrower,” “wider,” or the like. Save the chart that shows the words students use to describe angles during the activity to revisit in future lessons.

Here are the two sets of images for the activity and one set for the extension:

Set 1

Set 2

Set 3 (If time permits)

Engagement: Provide Access by Recruiting Interest. Synthesis: Optimize meaning and value. Ask, “How might thinking about angles be useful in our lives?” Consider making a connection to sports. For example, it might be easier to score in soccer if the ball is in front of the goal rather than off to the side, because of the angles involved. Show pictures if applicable and possible. (Consider drawing or labeling a picture in which the soccer ball is the vertex and the posts are points along the rays.)
Supports accessibility for: Conceptual Processing, Attention, Social-Emotional Functioning

### Required Materials

Materials to Gather

Materials to Copy

• Tricky Figures

### Required Preparation

• Create a set of cards (4 cards total) for each group of 2 from the blackline master.
• Each group of 2 needs 2 cards (sets 1 and 2). Additional cards (sets 3A and 3B) can be used for extension.

### Launch

• Groups of 2
• Read the instructions together as a class. Demonstrate or clarify the process as needed.

### Activity

• Give one partner the card for set 1. When the partners have discussed the drawings, give the other partner the card for set 2 and repeat the exercise.
• Provide one ruler or straightedge per student.
• 3–4 minutes per round

MLR2 Collect and Display

• Monitor for geometry terms from this section and phrases students use to describe the pace between the rays on each card. (spread apart, slide, moved over)
• Record students’ words and phrases on a visual display and update it throughout the lesson.
• If groups are ready for more, give them a third card to describe and draw.

### Student Facing

Work with a partner in this activity. Choose a role: A or B. Sit back to back, or use a divider to keep one person from seeing the other person’s work.

Partner A:

• Your teacher will give you a card. Don’t show it to your partner.
• Describe both images on the card—as clearly and precisely as possible—so that your partner can draw the same images.

Partner B:

• Your partner will describe two images. Listen carefully to the descriptions.
• Create the drawings as described. Follow the instructions as closely as possible.

1. When done, compare the drawings to the original images. Discuss:

• Which parts were accurate? Which were off?
• How could the descriptions be improved so the drawing could be more accurate?
2. Switch roles and repeat the exercise. Compare the drawings to the original images afterwards.

If you have time: Request two new cards from your teacher (one card at a time). Take turns describing and drawing the geometric figure on each card.

### Activity Synthesis

• “How are the two drawings on each card the same?” (They each have 2 rays. The rays start at the same point. One ray is pointing in the same direction in both drawings.)
• “How are they different?” (The rays are pointing in different directions on some cards. The rays are farther apart in some cards.)
• “How did you describe what you saw? What terms did you use to help you describe the directions of the rays?” (We tried to explain by describing the hands on a clock. We tried using words like north, south, east, and west. We described them in relation to vertical and horizontal.)
• As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.
• “Did anyone use the term ‘angle?’ Did anyone measure something or use measurements?”
• “The figures that you drew are angles. An angle is a figure that is made up of two rays that share the same endpoint.“
• “The point where the two rays meet is called the vertex of the angle.”

Display:

• “Where around us do we see angles?”

## Activity 2: Angles or Not Angles? (15 minutes)

### Narrative

In the previous activity, students learned what constitutes an angle. In this activity, they identify angles within geometric figures and explain their reasoning. Listen for the ways students show their understanding of rays even when they are not explicitly labeled in a given geometric figure.

This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing.

### Launch

• Groups of 2
• Read the prompt to students.

### Activity

MLR1 Stronger and Clearer Each Time

• 5 minutes: independent work time
• “Share your response to the first problem with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 3–5 minutes: partner discussion
• Repeat with 2–3 different partners.
• Monitor for students who hypothesize that there are no angles in figure B or C because no rays are drawn. Address this idea in the synthesis.
• “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time

### Student Facing

1. Decide if each figure shows at least one angle. Explain or show your reasoning for each.

2. Clare and Kiran are looking at this diagram. Clare says there are no angles because the rays do not meet at a point. Kiran says he sees two angles.

Do you agree with either of them? How many angles do you see?

### Activity Synthesis

• “How did you decide if there’s at least one angle in each figure? What did you look for?” (I looked for two lines or segments that intersect.)
• “How might you use the word ‘angle’ in our description of figure A? If so how?” (Two intersecting lines make 4 angles.)
• “In figure B, the sides of the quadrilateral are segments rather than rays. Does that mean there are no angles in the figure?” (No, there are angles. We can think of the segments as being parts of longer rays.)
• “Can you show the rays that make the angles in figure B?”

Display:

• “Can we show an angle in figure C by drawing rays, just as we have done with B?” (No, because one of the marks is not straight and can’t be a part of a ray.)
• “In Clare and Kiran’s argument, where are the two angles that Kiran saw?” (Kiran saw angles where each ray meets the line segment. The line segment is part of the rays that start at either of its endpoints.)

## Activity 3: Discover Angles (10 minutes)

### Narrative

In this activity, students identify and sketch angles in their environment—in the text, graphics, and shapes in their physical surroundings—and reinforce the idea of an angle as a figure made up of two rays that share an endpoint.

In future lessons, students will look more closely at the properties of angles and consider how they can be measured and whether the length of the segments that form them impacts the size of the angles.

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give each student a ruler or a straightedge.

### Activity

• 5 minutes: independent work time
• 2 minutes: partner discussion

### Student Facing

Here are two figures.

1. Find 2–3 angles in each figure. Draw pairs of rays to show the angles.
2. Sketch a part of your classroom that has 2–3 angles. Draw pairs of rays to show the angles.

### Student Response

Students may say they have identified an angle because they have found a vertex. Ask these students to explain what they mean and to describe the angle they see. Consider asking:

• “How do you see the rays that share this vertex? Can you draw them?”
• “Could more than one angle share a vertex? How could you use figure A from the previous activity to explain?”

### Activity Synthesis

• “How did you find the angles?” (We looked for intersections of segments. When the segments meet or cross, angles are formed.)
• “Where is the vertex of each angle you found?”

## Lesson Synthesis

### Lesson Synthesis

“Today we learned that an angle is a figure made up of two rays that share the same endpoint, and that the shared point is the vertex of the angle.”

”Use the words ‘sometimes’, ‘always’, or ‘never’ to respond to each statement about angles and lines:”

• “Intersecting lines form angles.” (Always, because they make rays that share the same endpoint.)
• “Parallel lines can form angles.” (Never, because they will never meet or share a point.)
• “Angles can be formed by curves.” (Never, because a ray is a part of a line, which is always straight.)

“Take 1–2 minutes to add the new words from today’s lesson to your word wall. Share your new entries with a neighbor and add any new ideas you learn from your conversation.”